BRST quantization
[1]} is an important and powerful technique to deal with a system with constraints [2]}, [3]}, [4]}. It enlarges the phase space of a gauge theory and restores the symmetry of the gauge fixed action in the extended phase space keeping the physical contents of the theory unchanged. BRST symmetry plays a very important role in renormalizing spontaneously broken theories, like standard model and hence it is extremely important to investigate it for different systems. To the best of our knowledge BRST formulation for a non-relativistic particle constrained to a curved surface embedded in the higher dimensional Euclidean space has not been developed yet. This motivates us in the study of BRST symmetry for this system. We study non-relativistic particle constrained to a curved surface embedded in the higher dimensional Euclidean space following the technique of Dirac’s constraints analysis [2]}, [4]}, [3]}. The system is shown to contain second-class constraints. We will apply Batalin-Fradkin-Fradkina-Tyutin (BFFT) method to convert these second class constraints to first class constraints [8]}, [9]}. We further develop the BRST charge and symmetry for this BFFT extended system using BFV (Batalin-Fradkin-Vilkovisky) formulation from the constraints of the theory [3]}, [11]}. In the limit \(L\rightarrow (N-1)\)  the results have been obtained which are shown to be consistent with the previous result [12]}. The possible method of constructing nilpotent BRST charge in the operator form using the mode expansion of the fields [13]} has been suggested and a short discussion on possible correction in Hamiltonian due to BRST quantization and its comparison with Dirac quantization is given [14]}. The result has been verified using a simple example of particle on torus knot[15]}, [16]} by calculating the first class constraints, Hamiltonian BRST charge and symmetry for it. At the end we will construct BRST transformation for this system using Batalin-Vilkovisky (BV) quantization method [17]}, [18]}, [19]}.
